The Mathematical Work of the 2002 Fields Medalists, Volume 50

The Mathematical Work of the 2002 Fields Medalists Eric M. Friedlander, Michael Rapoport, and Andrei Suslin

The Work of Laurent Lafforgue hand and of the idèle class group A×/F × on the Michael Rapoport other hand. This is the reciprocity law of T. Takagi Laurent Lafforgue was awarded the Fields Medal for and E. Artin established in the 1920s as a far-reach- his proof of the Langlands correspondence for the ing generalization of the quadratic reciprocity law general linear groups GLr over function fields of going back to L. Euler. At the end of the 1960s, positive characteristic. His approach to this prob- R. Langlands proposed a nonabelian generaliza- lem follows the basic strategy introduced twenty- tion of this reciprocity law. It conjecturally relates five years ago by V. Drinfeld in his proof for GL2. the irreducible representations of rank r of Gal(F/F¯ ) Already Drinfeld’s proof is extremely difficult. Laf- (or, more generally, of the hypothetical motivic forgue’s proof is a real tour de force, taking up as Galois group of F) with cuspidal automorphic rep- it does several hundred pages of highly condensed resentations of GL (A). In fact, this conjecture is reasoning. By his achievement Lafforgue has proved r part of an even grander panorama of Langlands (the himself a mathematician of remarkable strength functoriality principle), in which homomorphisms and perseverance. In this brief report I will sketch the background between L-groups of reductive groups over F induce of Lafforgue’s result, state his theorems, and then relations between the automorphic representations mention some ingredients of his proof. The final on the corresponding groups. These hypothetical passages are devoted to the human factor. reciprocity laws would imply famous conjectures, The Background such as the Artin conjecture on the holomorphy of The background of Lafforgue’s theorem is the web L-functions of irreducible Galois representations of conjectures known as the Langlands philosophy, and the Ramanujan-Petersson conjecture on the

which is a far-reaching generalization of class field Hecke eigenvalues of cusp forms for GLr. theory. Let F be a global field, i.e., either a finite In the number field case, deep results along extension of Q (the number field case) or a finite ex- these lines have been obtained for groups of small tension of Fp(t) where Fp is the finite field with p rank, such as GL2, by Langlands himself and by elements (the function field case). Let A be the adèle many others. And such results have already had ring of F. spectacular applications, such as in the proof of Global class field theory may be formulated as giving a bijection between the sets of characters of Fermat’s last theorem. However, the proof of the finite order of the Galois group Gal(F/F¯ ) on the one Langlands correspondence in any generality in the number field case seems out of reach at the Michael Rapoport is professor of mathematics at the Uni- present time. Lafforgue’s result, which concerns versität Köln. His email address is rapoport@math. the function field case, is the first general non- uni-koeln.de. abelian reciprocity law.

212 NOTICES OF THE AMS VOLUME 50, NUMBER 2 Lafforgue’s Theorem product formula for ε-factors of G. Laumon), it all From now on let F denote a function field of char- boiled down to proving the existence of the map acteristic p. We also fix an auxiliary prime number π σ (π) with the required properties. This is ex- = p. For a positive integer r let Gr be the set of actly what Lafforgue did. equivalence classes of irreducible -adic repre- Before spending a few words on his proof, let us sentations of dimension r of Gal(F/F¯ ). For each consider the question, What is it good for? The an- σ ∈Gr , A. Grothendieck defined its L-function swer is that neither of the sets Gr and Ar is simpler L(σ,s), which is a rational function in p−s satisfy- than the other in every aspect but that Theorem 1 can ing a functional equation of the form be used to transfer available information in either L(σ,s) = ε(σ,s) · L(σ ∨, 1 − s) , where ε(σ,s) is a direction. Theorem 3 is an instance where informa- −s ∨ monomial in p and σ denotes the contragredi- tion available on Ar implies results on Gr. In the ent representation. The L-function is an Euler prod- other direction, Theorem 1 permits one to use uct, L(σ,s) = ΠxLx(σ,s), over all places x of F, and constructions available on Gr to prove various in- for a place x of degree deg(x), where σ is unrami- stances of Langlands functoriality for Ar, such as fied, we have the existence of tensor products, of base change, r and of automorphic induction. 1 Lx(σ,s) = . About the Proof 1 − z p−s deg(x) i=1 i The strategy of constructing the map π σ (π) is due to Drinfeld and is inspired by the work of Here z1,...,zr are the Frobenius eigenvalues of σ at x. Y. Ihara, Langlands, and others in the theory of Shimura varieties. It consists in analyzing the -adic Let Ar be the set of equivalence classes of cuspidal representations of GL (A). For each π ∈A, cohomology of the algebraic stack Shtr,∅ over r r × R. Godement and H. Jacquet defined its L-function Spec F Spec F parametrizing shtukas of rank r = L(π,s) with properties similar to those of the above or the algebraic stack Shtr lim← Shtr,N parame- L-functions. The Euler factor at a place x where π trizing shtukas of rank r equipped with a com- is unramified is given as patible system of level structures for all levels N. The latter cohomology module is equipped with an r 1 action of GL (A) × Gal(F/F¯ ) × Gal(F/F¯ ) , and the L (π,s) = , r x − −s deg(x) aim is to isolate inside it a subquotient of the form = 1 zi p i 1 ⊗ ⊗ ∨ where z1,...,zr are the Hecke eigenvalues of π π σ (π) σ (π) at x. The main result of Lafforgue consists of the π∈Ar following theorems. by comparing the Grothendieck-Lefschetz fixed- Theorem 1 (the Langlands conjecture). There is point formula and the Arthur-Selberg trace for- a bijection π σ (π) between Ar and Gr, charac- mula. The essential difficulty is that, in contrast to terized by the fact that Lx(π,s) = Lx(σ (π),s) for the case of Shimura varieties, the moduli stack every place x of F. Shtr is not of finite type, not even at any finite level N. To understand why, recall that a shtuka Theorem 2 (the Ramanujan-Petersson conjec- of rank r is a vector bundle of rank r on X with additional structure (essentially a meromorphic ture). Let π ∈Ar with central character of finite order. Then for every place x of F where π is un- descent datum under Frobenius). Here X is the smooth irreducible projective curve over Fp with ramified, the Hecke eigenvalues z1,...,zr ∈ C are all of absolute value 1. function field F. And, just as the moduli stack of vector bundles of rank r on X is not of finite type, Sht ∅ Sht Theorem 3 (the Deligne conjecture). Let σ ∈G neither are the stacks r, and r,N. r To deal with this difficulty, Lafforgue introduces with determinant character of finite order. Then σ ≤ ≤ the open substacks Sht P and Sht P of shtukas is pure of weight 0, i.e., for any place x of F where r,∅ r,N where the Harder-Narasimhan polygon is bounded σ is unramified, the images of the Frobenius eigen- by P. These substacks are of finite type, and their values z ,...,z under any embedding of Q¯ into 1 n union is the whole space. The trouble is that they C are of absolute value 1. are not stable under the Hecke correspondences. Here Theorems 2 and 3 are consequences of Therefore Lafforgue constructs in the case without ≤P Theorem 1 through P. Deligne’s purity theorem level structure a smooth compactification Shtr,∅ of ≤P and the estimate on Hecke eigenvalues of Jacquet Shtr,∅ with a normal crossing divisor at infinity and and J. Shalika. Theorem 1 itself is proved by in- extends the Hecke correspondences to it by duction on r (Deligne recursion principle). After simple normalization. He then applies the Grothen- what was known before (in addition to the func- dieck-Lefschetz fixed-point formula to these tional equations, essentially the converse theo- correspondences. However, only a part of this for- rems of A. Weil and I. Piatetskii-Shapiro and the mula can be determined explicitly, and therefore

FEBRUARY 2003 NOTICES OF THE AMS 213 this seems a pointless exercise. Lafforgue circum- Since 2000 he has been a professor at the Institut vents this problem by isolating the r-essential part des Hautes Études Scientifiques. ≤P of the cohomology of Shtr,∅ and by showing that Further Information the remainder, both the difference between the For excellent overviews of Lafforgue’s proof, see ≤P cohomology of Shtr,∅ and of Shtr,∅ and the coho- Laumon’s Bourbaki seminar No. 873, March 2000, mol ≤P ogy of the boundary of Shtr,∅ , is r-negligible. which also contains an annotated bibliography, Here the work of R. Pink on Deligne’s conjecture and Lafforgue’s notes of a course at the Tata In- on the Grothendieck-Lefschetz formula enters in stitute (http://www.ihes.fr/PREPRINTS/ a decisive way. In the case where a level structure M02/Resu/resu-M02-45.html). is imposed, Lafforgue manages to push through his method by constructing a partial compactification ≤P of Shtr,N which is smooth with a normal crossing The Work of Vladimir Voevodsky divisor at infinity and which is stable under the Eric M. Friedlander and Andrei Suslin Hecke correspondences and by supplementing In 1982 Alexander Beilinson stated conjectures Pink’s theorem by K. Fujiwara’s theorem. which crystallized a vision of the relationship be- The Months of Suspense tween algebraic K-theory and “integral motivic co- Lafforgue’s first attempt at a proof of Theorem 1 homology theory” of algebraic varieties over a field ≤P used a compactification of Shtr,N. His construc- F and between mod- algebraic K-theory and tion was based on the compactifications of the mod- étale cohomology, where is a prime in- = n+1 quotient spaces Xr,n (PGLr ) /PGLr that he had vertible in F [Be]. These conjectures, more detailed defined in earlier work, generalizing the case n = 1 and specific than the earlier “Quillen-Lichtenbaum due to C. De Concini and C. Procesi. In June 2000, Conjecture”, provided a challenging program that while lecturing on his proof, Lafforgue discovered explains a great deal about Quillen’s algebraic that, contrary to what he had claimed, these com- K-theory [Q]. Although one of Beilinson’s conjec- pactifications of Xr,n, and hence also the corre- tures remains unsolved (and may well prove to be ≤P sponding compactifications of Shtr,N, are not false in general), Vladimir Voevodsky has made smooth in general. He was not even able to resolve great strides in completing Beilinson’s program. their singularities. During two months of suspense Voevodsky’s achievements are remarkable. First, in the summer of 2000, Lafforgue managed to fill he has developed a general homotopy theory for the gap by finding the above-mentioned partial algebraic varieties. Second, as part of this general ≤P compactifications of Shtr,N and was able to finesse theory, he has formulated what appears to be the the proof of Theorem 1 from them. Thus in the end, “correct” motivic cohomology theory and verified the modified argument is simpler than the origi- many of its remarkable properties. Third, as an ap- nal attempt. plication of this general approach, he has proved Even though Lafforgue’s compactifications of Xr,n a long-standing conjecture of John Milnor relating are not used in the final proof, they are fascinating the Milnor K-theory of a field to its étale coho- objects in themselves, with close relations to such di- mology (and to quadratic forms over the field). verse geometric objects as configuration spaces of We provide below a brief sketch of Voevodsky’s matroids, thin Schubert cells, stable degeneration contributions, proceeding from the specific Mil- of n-pointed projective lines, and local models of nor Conjectures to the more general theory. Shimura varieties. It turns out that these compacti- Milnor Conjecture = fications are smooth for n 1 (respectively toroidal Given a field F, we define (following Milnor) the = for n 2) and arbitrary r (De Concini and Procesi, re- graded ring KM (F) to be the tensor algebra over the = ∗ spectively Lafforgue) and for r 2 and arbitrary n integers Z of the group F ∗ of nonzero elements in (G. Faltings), but can have arbitrarily bad singulari- F, modulo the ideal generated by the Steinberg re- ties in general (N. Mnev). These compactifications lations in F ∗ ⊗ F ∗ (so that {a, 1 − a}, the image of constitute a new field of investigation, taken up by ⊗ − ∈ M = a (1 a), is set equal to 0 K2 (F) for all a 1 Lafforgue in a 265-page preprint (http://www.ihes. ∈ ∗ M = = Z M = = F ). Thus, K0 (F) K0(F) , K1 (F) K1(F) fr/PREPRINTS/M02/Resu/resu-M02-31.html). F ∗, and (thanks to a theorem of Matsumoto) Biographical Data Laurent Lafforgue was born in 1966. He was a stu- Eric M. Friedlander is professor of mathematics at North- dent at the École Normale Supérieure (1986–1990) western University. His work was partially supported by before entering the Centre National des Recherches NSF grant DMS-9988130 and by the NSA. His email ad- Scientifiques in 1990. His academic teacher is dress is [email protected]. Gérard Laumon, with whom he obtained his thèse Andrei Suslin is professor of mathematics at Northwest- at the Université de Paris-Sud in 1994. It is in the ern University. His work was partially supported by NSF- famous Bâtiment de Mathématique (“le 425”) on the grant DMS-0100586. His email address is suslin@ Orsay campus that Lafforgue worked out his proof. math.nwu.edu.

214 NOTICES OF THE AMS VOLUME 50, NUMBER 2 M = = ∗ ⊗ ∗ ⊗ − = ∈ ∗ K2 (F) K2(F) F F /(a (1 a); a 1 F ). providing a substitute for singular cohomology M mod- whenever does not equal the characteris- By definition, the n-th Milnor K-group Kn (F) is generated by products of elements in degree 1 (so- tic of F. called symbols {a1,...,an}), thereby being much Beilinson conjectured the existence of motivic Z simpler than Quillen’s K-group Kn(F). complexes (q) leading to bigraded motivic coho- The following theorem, conjectured by Milnor mology groups Hp(X,Z(q)) which should relate to [Mi], has been proved by Voevodsky. the algebraic K-theory of a smooth variety X over a field F. (This relationship has recently seen various Theorem 1. [V4] Let F be a field of characteristic proofs, beginning with work of Bloch-Lichtenbaum, different from 2. Then the natural map then Friedlander-Suslin, Levine, and finally Grayson- Suslin.) Moreover, he conjectured that these com- KM (F) ⊗ Z/2Z → H∗(F,Z/2Z) ∗ plexes, when reduced modulo for invertible in F, should be related in a completely precise way to étale is an isomorphism of graded rings. In particular, cohomology. for every n>0, the Galois cohomology group In [Bl] Spencer Bloch produced an ingenious con- n Z Z H (F, /2 ) is generated by n-fold cup products of struction yielding “higher Chow groups”, which are cohomology classes of degree 1. excellent candidates for the cohomology of Beilin- One can marvel at this result from many points son’s conjectured motivic complexes. Somewhat of view. For example, it says there is something very later, Suslin introduced algebraic singular complexes special about graded rings that arise as Galois co- which led Suslin and Voevodsky to an alternate homology of fields. algebraic model of singular cohomology with Voevodsky, in collaboration with D. Orlov and mod- coefficients [SV1]. A major achievement of A. Vishik, has also proved the following closely re- Voevodsky has been to formulate a natural category lated theorem conjectured by Milnor [Mi]. The (of abelian presheaves with transfers on the big Grothendieck-Witt ring GW(F) has an underlying Nisnevich site of all smooth varieties X over F) which abelian group given by isomorphism classes of in conjunction with the algebraic singular complex finite-dimensional F-vector spaces equipped with a construction leads to a good formulation of motivic quadratic form and ring structure given by tensor complexes Z(n) developed by Suslin and Voevodsky product. The Witt ring W (F) is obtained by dividing [SV2]. Recently Voevodsky has proved that the GW(F) by the ideal generated by the 2-dimensional resulting cohomology groups are isomorphic to F-vector space with the hyperbolic quadratic form. Bloch’s higher Chow groups [V2]. The Witt ring W (F) admits an augmentation map Voevodsky’s formulation of motivic cohomology W (F) → Z/2Z induced by the rank map GW(F) → Z. is crucial, for it has led Voevodsky to prove various Milnor investigated the successive quotients In/In+1 , important properties essential for his proof of the where I ⊂ W (F) denotes the kernel of the augmen- Milnor Conjectures. We mention perhaps the most tation map, and the resulting associated graded ring important general theorem concerning motivic n n+1 cohomology, a theorem for which Voevodsky has gr∗(W (F)) =⊕n≥0I /I . recently found a most elegant proof. Theorem 2. [OVV] Let F be a field of characteristic different from 2. Then the natural homomor- Theorem 3. [FV], [V5] Let X be a smooth variety ⊂ phism of graded rings K∗(F) ⊗ Z/2Z → gr∗(W (F)) is over a field F, and let Y X be a smooth closed an isomorphism. subvariety everywhere of codimension d. Then there is a canonical Gysin isomorphism Theorems 1 and 2 show that the three graded p − M ∗ Z = p 2d Z − rings K∗ (F) ⊗ Z/2Z, gr∗(W (F)), and H (F,Z/2Z), HY (X, (n)) H (Y, (n d)). which arise in very different manners, are canon- Operations in Motivic Cohomology and the Proof ically isomorphic for any field of characteristic of the Milnor Conjecture different from 2. Steenrod operations in mod- motivic cohomology Motivic Cohomology are at the heart of Voevodsky’s proof of the Milnor In algebraic topology, singular cohomology with Conjecture. The construction of these operations integral coefficients has many good properties and given in [V3] turned out to be much more subtle than satisfies a tight relationship with topological the topological counterpart, and their properties K-theory. One has known for many years that one in the special case = 2 differ somewhat from the cannot define “algebraically” the integral (or even ra- corresponding properties of Steenrod operations in tional) singular cohomology of complex algebraic va- topological singular cohomology. rieties. On the other hand, étale cohomology with Following the approach for n = 2 introduced by mod- coefficients, as developed by Alexander A. Merkurjev and Suslin, Voevodsky investigated Grothendieck and Michael Artin, when applied to va- what happens when one splits a symbol { }∈ M ⊗ Z rieties over a field F , succeeds admirably in a1,...,an Kn / . For any such symbol,

FEBRUARY 2003 NOTICES OF THE AMS 215 there exists a universal splitting variety Xa. Earlier The Homotopy Category of Schemes work of Suslin and Voevodsky [SV2], together with Beginning with his Harvard Ph.D. thesis, Voevod- several ingenious arguments introduced by Vo- sky has had the goal of creating a homotopy the- evodsky, reduces the proof of the Milnor Conjec- ory for algebraic varieties amenable to calculations ture (and the more general Bloch-Kato Conjecture) as in algebraic topology. Much of this homotopy to a specific vanishing calculation in motivic co- ∨ theory, both stable and unstable, has been devel- homology associated to X : Hn+1(C(X ), Z (n)) = 0, oped in collaboration with Fabien Morel (see [MV]). ∨ a a ( ) In this abstract context one can realize motivic co- where∨ C(Xa) is a simplicial scheme with = n+1 homology and algebraic K-theory as representable Cn(Xa) Xa and all face (respectively, degener- acy) maps given by projections (respectively, di- functors. One can view motivic Steenrod opera- agonal embeddings). tions as a special case of operations on “general- Associated to Voevodsky’s Steenrod operations ized stable cohomology theories”. When Voevod- are the corresponding Milnor operations sky localizes to obtain a homotopy category of + i − schemes, he is forcing “homotopy invariance”, the Q : H p(−, Z/ (q)) → H p 2 1(−, Z/ (q + i − 1)). i property that a homotopy type (e.g., represented As in algebraic topology, these operations satisfy by a scheme) is viewed as equivalent to the prod- 2 = the property Qi 0, so motivic cohomology pro- uct of itself and the affine line. A key insight of Vo- vided with the operator Qi forms a complex, whose evodsky is that there are two types of “circles” in homology is known as Margolis homology. As a next algebraic geometry determining two types of sus- step, Voevodsky proved the following remarkable pension. One type of circle arises when one con- theorem concerning the vanishing of Margolis siders an affine nodal curve (tracing out a curve homology. which crosses itself) and the other when one con- Theorem 4. [V3] Let X be a smooth projective va- siders the punctured affine line (which is simply riety over a field k of characteristic different from real 2-space minus the origin in the special case the . Assume that there exists a morphism Y → X ground field is the complex numbers). from a smooth projective variety Y of dimension Although Voevodsky has given us a relatively m − 1 to X, and further assume that the charac- streamlined proof of the Milnor Conjecture which does not rely on this homotopy category, his orig- teristic number deg(s m−1(Y )) is not congruent to 0 modulo 2 . Then all of the Margolis homology inal conception of the proof relied heavily on such groups of the simplicial sheaf C(X) (the unreduced a homotopy-theoretic point of view. Moreover, the ∨ suspension of C(X)) corresponding to the opera- expected proof of the odd-prime analogue of the Milnor Conjecture (the so-called “Bloch-Kato Con- tion Qm are 0. jecture”) fully utilizes this appealing formalism. = For 2, one can represent the splitting vari- We conclude by observing that not only has Vo- Q ety Xa by the quadric q defined by the Pfister evodsky’s work much influenced how algebraic = ⊥ − neighbor q a1,...,an−1 an of a1,..., geometers are approaching certain classical ques- an , which is a smooth projective variety of di- tions but also algebraic topologists have begun to n−1 − mension 2 1 and satisfies the condition that produce considerable foundational material in this deg(s n− (X )) is not divisible by 4. We now apply 2 1 a new homotopy theory. = =Q Theorem 4 with X Xa and Y a1,...,am−1⊥−an for 1 ≤ m ≤ n to conclude the vanishing of the References Margolis homology of C(X ) with respect to each a [Be] A. BEILINSON, letter to C. Soulé, 1982. of the operations Q1,...,Qn. [Bl] S. BLOCH, Algebraic cycles and higher K-theory, Adv. Using dimension considerations, Voevodsky ob- in Math. 61 (1986), 267–304. serves that the operation Qn−2 ◦···◦Q1 and its in- [FV] E. FRIEDLANDER and V. VOEVODSKY, Bivariant cycle co- tegral counterpart homology, Cycles, transfers, and Motivic Homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. ◦···◦ n+2 Z Qn−2 Q1 : H (C(X), (2)(n)) Press, Princeton, NJ, 2000, pp. 138–87. 2n n−1 [Mi] J. MILNOR, Algebraic K-theory and quadratic forms, → H (C(X), Z(2)(2 )) Invent. Math. 9 (1970), 318–44. A1 are monomorphisms. This reduces the proof of [MV] F. MOREL and V. VOEVODSKY, -homotopy theory of ∨ schemes, Inst. Hautes Études Sci. Publ. Math. No. 90 the vanishing of Hn+1(C(X), Z (n)) to the vanish- ∨ (2) (1999), 45–143 (2001). 2n−1 n−1 ing of the group H (C(X), Z(2)(2 )). Fortunately [OVV] D. ORLOV, A. VISHIK, and V. VOEVODSKY, An exact se- the latter group is much easier to understand: quence for Milnor’s K-theory with applications to qua- Using [R1], Voevodsky shows that this group is dratic forms, 2000 eprint, http://www.math. closely related to the group of 0-dimensional K1- uiuc.edu/K-theory/0454/, also arXiv:math. cycles studied closely by M. Rost in [R2]. To finish AG/0101023. the proof, Voevodsky then applies the main theo- [Q] D. QUILLEN, Higher Algebraic K-Theory: I, Lecture Notes rem of [R2]. in Math., vol. 341, Springer, Berlin, 1973, pp. 85–147.

216 NOTICES OF THE AMS VOLUME 50, NUMBER 2 [R1] M. ROST, The motive of a Pfister form, 1998 eprint, http://www.math.ohio-state.edu/~rost/ motive.html. [R2] ——— , Some new results on the Chow groups of quadrics, 1990 eprint, http://www.math.ohio- state.edu/~rost/chowqudr.html. [SV1] A. SUSLIN and V. VOEVODSKY, Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), 61–94. [SV2] ———, Bloch-Kato conjecture and motivic cohomology with finite coefficients, The Arithmetic and Geom- etry of Algebraic Cycles, NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer, Dordrecht, 2000, pp. 117–89. [V1]V. VOEVODSKY, Cohomological theory of presheaves with transfers, Cycles, Transfers, and Motivic Homol- ogy Theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 87–137. [V2] ——— , Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. no. 7 (2002), 351–5. [V3] ——— , Reduced power operations in motivic cohomology, 2001 eprint, http://www.math.uiuc.edu/ K-theory/0487/, also arXiv:math.AG/0107109. [V4] ——— , On 2-torsion in motivic cohomology, 2001 eprint, http://www.math.uiuc.edu/K-theory/ 0502/, also arXiv:math.AG/0107110. [V5] ———, Cancellation theorem, 2002 eprint, http:// www.math.uiuc.edu/K-theory/0541/, also arXiv:math.AG/0202012.

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